Question: Solve for $X$. $\left[\begin{array}{rr}8 & 12 \\ -4 & 2 \\16 &5\end{array}\right]-X=\left[\begin{array}{rr}6 & 6 \\ 0 & 9 \\1 &2\end{array}\right] $ $X=$
Answer: The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $\left[\begin{array}{rr}8 & 12 \\ -4 & 2 \\16 &5\end{array}\right]-X=\left[\begin{array}{rr}6 & 6 \\ 0 & 9 \\1 &2\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}8 & 12 \\ -4 & 2 \\16 &5\end{array}\right] ~~~~~~~~~ B = \left[\begin{array}{rr}6 & 6 \\ 0 & 9 \\1 &2\end{array}\right]$ Then we can rewrite the equation as follows. $A-X=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}A-X&=B\\\\ A&=B+X\\\\ X&=A-B \end{aligned}$ Finding $X$ We found that $X=A-B$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=A-B \\\\&=\left[\begin{array}{rr}8 & 12 \\ -4 & 2 \\16 &5\end{array}\right]-\left[\begin{array}{rr}6 & 6 \\ 0 & 9 \\1 &2\end{array}\right] \\\\\\&=\left[\begin{array}{rr}(8-6) & (12-6) \\ (-4-0) & (2-9) \\(16-1) &(5-2) \end{array}\right] \\\\\\&=\left[\begin{array}{rr}2 & 6 \\ -4 & -7 \\15 &3\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}2 & 6 \\ -4 & -7 \\15 &3\end{array}\right]$